OK, folks. I contacted Paragon & have a solution to my problem. Phil was (of course) on the right track. Here's what to do:
1. Remove control panel. This is the part where the buttons are for programming. It is held on by 4 screws + washers (phillips).
2. Gently pull control panel assembly out of the case, there are many wires attached.
3. Remove circuit board from panel cover. 4 screws + insulating washers on inside 4 corners.
4. The beeping thing (piezo) is round with a hole in the middle--maybe 3/4 inch tall, the diameter of a nickel. Tape over the hole.
5. Replace everything. taking care to gently push wires inside the case.
Voila! The beep is gentle & won't alarm the neighbors, who tend to think any beeping sound is a smoke alarm.
This is for my Paragon Caldera with Sentry 4.0 controller. Other brands & models may be different. YMMV.
No skinned knuckles, no bleeding, not even very much swearing involved. Only casualty is a hole in my favorite t shirt that snagged on the bead door while I was working.